In this note, we illustrate the computation of the approximation of the supply curves using a one-step basis. We derive the expression for the L2 approximation and propose a procedure for the selection of nodes of the approximation. We illustrate the use of this approach with three large sets of bid curves from European electricity markets.
{"title":"Approximation of supply curves","authors":"Andres M. Alonso, Zehang Li","doi":"arxiv-2311.10738","DOIUrl":"https://doi.org/arxiv-2311.10738","url":null,"abstract":"In this note, we illustrate the computation of the approximation of the\u0000supply curves using a one-step basis. We derive the expression for the L2\u0000approximation and propose a procedure for the selection of nodes of the\u0000approximation. We illustrate the use of this approach with three large sets of\u0000bid curves from European electricity markets.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522560","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Options, serving as a crucial financial instrument, are used by investors to manage and mitigate their investment risks within the securities market. Precisely predicting the present price of an option enables investors to make informed and efficient decisions. In this paper, we propose a machine learning method for forecasting the prices of SPY (ETF) option based on gated recurrent unit (GRU) and self-attention mechanism. We first partitioned the raw dataset into 15 subsets according to moneyness and days to maturity criteria. For each subset, we matched the corresponding U.S. government bond rates and Implied Volatility Indices. This segmentation allows for a more insightful exploration of the impacts of risk-free rates and underlying volatility on option pricing. Next, we built four different machine learning models, including multilayer perceptron (MLP), long short-term memory (LSTM), self-attention LSTM, and self-attention GRU in comparison to the traditional binomial model. The empirical result shows that self-attention GRU with historical data outperforms other models due to its ability to capture complex temporal dependencies and leverage the contextual information embedded in the historical data. Finally, in order to unveil the "black box" of artificial intelligence, we employed the SHapley Additive exPlanations (SHAP) method to interpret and analyze the prediction results of the self-attention GRU model with historical data. This provides insights into the significance and contributions of different input features on the pricing of American-style options.
期权作为一种重要的金融工具，被投资者用来管理和减轻证券市场上的投资风险。准确预测期权的当前价格使投资者能够做出明智而有效的决策。本文提出了一种基于门控递归单元(GRU)和自关注机制的SPY (ETF)期权价格预测机器学习方法。我们首先根据钱数和到期日标准将原始数据集划分为15个子集。对于每个子集，我们匹配相应的美国政府债券利率和隐含波动率指数。这种分割允许对无风险利率和潜在波动率对期权定价的影响进行更有见地的探索。接下来，我们建立了四种不同的机器学习模型，包括多层感知器(MLP)、长短期记忆(LSTM)、自注意LSTM和自注意GRU，并与传统的二项模型进行了比较。实证结果表明，具有历史数据的自关注GRU优于其他模型，因为它能够捕获复杂的时间依赖性并利用嵌入在历史数据中的上下文信息。最后，为了揭开人工智能的“黑盒子”，我们采用SHAP (the hapley Additive explanatory)方法，结合历史数据对自关注GRU模型的预测结果进行了解释和分析。这就揭示了不同输入特征对美式期权定价的意义和贡献。
{"title":"American Option Pricing using Self-Attention GRU and Shapley Value Interpretation","authors":"Yanhui Shen","doi":"arxiv-2310.12500","DOIUrl":"https://doi.org/arxiv-2310.12500","url":null,"abstract":"Options, serving as a crucial financial instrument, are used by investors to\u0000manage and mitigate their investment risks within the securities market.\u0000Precisely predicting the present price of an option enables investors to make\u0000informed and efficient decisions. In this paper, we propose a machine learning\u0000method for forecasting the prices of SPY (ETF) option based on gated recurrent\u0000unit (GRU) and self-attention mechanism. We first partitioned the raw dataset\u0000into 15 subsets according to moneyness and days to maturity criteria. For each\u0000subset, we matched the corresponding U.S. government bond rates and Implied\u0000Volatility Indices. This segmentation allows for a more insightful exploration\u0000of the impacts of risk-free rates and underlying volatility on option pricing.\u0000Next, we built four different machine learning models, including multilayer\u0000perceptron (MLP), long short-term memory (LSTM), self-attention LSTM, and\u0000self-attention GRU in comparison to the traditional binomial model. The\u0000empirical result shows that self-attention GRU with historical data outperforms\u0000other models due to its ability to capture complex temporal dependencies and\u0000leverage the contextual information embedded in the historical data. Finally,\u0000in order to unveil the \"black box\" of artificial intelligence, we employed the\u0000SHapley Additive exPlanations (SHAP) method to interpret and analyze the\u0000prediction results of the self-attention GRU model with historical data. This\u0000provides insights into the significance and contributions of different input\u0000features on the pricing of American-style options.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Perpetual futures are contracts without expiration date in which the anchoring of the futures price to the spot price is ensured by periodic funding payments from long to short. We derive explicit expressions for the no-arbitrage price of various perpetual contracts, including linear, inverse, and quantos futures in both discrete and continuous-time. In particular, we show that the futures price is given by the risk-neutral expectation of the spot sampled at a random time that reflects the intensity of the price anchoring. Furthermore, we identify funding specifications that guarantee the coincidence of futures and spot prices, and show that for such specifications perpetual futures contracts can be replicated by dynamic trading in primitive securities.
{"title":"Perpetual Futures Pricing","authors":"Damien Ackerer, Julien Hugonnier, Urban Jermann","doi":"arxiv-2310.11771","DOIUrl":"https://doi.org/arxiv-2310.11771","url":null,"abstract":"Perpetual futures are contracts without expiration date in which the\u0000anchoring of the futures price to the spot price is ensured by periodic funding\u0000payments from long to short. We derive explicit expressions for the\u0000no-arbitrage price of various perpetual contracts, including linear, inverse,\u0000and quantos futures in both discrete and continuous-time. In particular, we\u0000show that the futures price is given by the risk-neutral expectation of the\u0000spot sampled at a random time that reflects the intensity of the price\u0000anchoring. Furthermore, we identify funding specifications that guarantee the\u0000coincidence of futures and spot prices, and show that for such specifications\u0000perpetual futures contracts can be replicated by dynamic trading in primitive\u0000securities.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Cryptocurrencies and Bitcoin, in particular, are prone to wild swings resulting in frequent jumps in prices, making them historically popular for traders to speculate. A better understanding of these fluctuations can greatly benefit crypto investors by allowing them to make informed decisions. It is claimed in recent literature that Bitcoin price is influenced by sentiment about the Bitcoin system. Transaction, as well as the popularity, have shown positive evidence as potential drivers of Bitcoin price. This study considers a bivariate jump-diffusion model to describe Bitcoin price dynamics and the number of Google searches affecting the price, representing a sentiment indicator. We obtain a closed formula for the Bitcoin price and derive the Black-Scholes equation for Bitcoin options. We first solve the corresponding Bitcoin option partial differential equation for the pricing process by introducing artificial neural networks and incorporating multi-layer perceptron techniques. The prediction performance and the model validation using various high-volatile stocks were assessed.
{"title":"Neural Network for valuing Bitcoin options under jump-diffusion and market sentiment model","authors":"Edson Pindza, Jules Clement Mba, Sutene Mwambi, Nneka Umeorah","doi":"arxiv-2310.09622","DOIUrl":"https://doi.org/arxiv-2310.09622","url":null,"abstract":"Cryptocurrencies and Bitcoin, in particular, are prone to wild swings\u0000resulting in frequent jumps in prices, making them historically popular for\u0000traders to speculate. A better understanding of these fluctuations can greatly\u0000benefit crypto investors by allowing them to make informed decisions. It is\u0000claimed in recent literature that Bitcoin price is influenced by sentiment\u0000about the Bitcoin system. Transaction, as well as the popularity, have shown\u0000positive evidence as potential drivers of Bitcoin price. This study considers a\u0000bivariate jump-diffusion model to describe Bitcoin price dynamics and the\u0000number of Google searches affecting the price, representing a sentiment\u0000indicator. We obtain a closed formula for the Bitcoin price and derive the\u0000Black-Scholes equation for Bitcoin options. We first solve the corresponding\u0000Bitcoin option partial differential equation for the pricing process by\u0000introducing artificial neural networks and incorporating multi-layer perceptron\u0000techniques. The prediction performance and the model validation using various\u0000high-volatile stocks were assessed.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
At the peak of the tech bubble, only 0.57% of market valuation comes from dividends in the next year. Taking the ratio of total market value to the value of one-year dividends, we obtain a valuation-based duration of 175 years. In contrast, at the height of the global financial crisis, more than 2.2% of market value is from dividends in the next year, implying a duration of 46 years. What drives valuation duration? We find that market participants have limited information about cash flow beyond one year. Therefore, an increase in valuation duration is due to a decrease in the discount rate rather than good news about long-term growth. Accordingly, valuation duration negatively predicts annual market return with an out-of-sample R2 of 15%, robustly outperforming other predictors in the literature. While the price-dividend ratio reflects the overall valuation level, our valuation-based measure of duration captures the slope of the valuation term structure. We show that valuation duration, as a discount rate proxy, is a critical state variable that augments the price-dividend ratio in spanning the (latent) state space for stock-market dynamics.
{"title":"Valuation Duration of the Stock Market","authors":"Ye Li, Chen Wang","doi":"arxiv-2310.07110","DOIUrl":"https://doi.org/arxiv-2310.07110","url":null,"abstract":"At the peak of the tech bubble, only 0.57% of market valuation comes from\u0000dividends in the next year. Taking the ratio of total market value to the value\u0000of one-year dividends, we obtain a valuation-based duration of 175 years. In\u0000contrast, at the height of the global financial crisis, more than 2.2% of\u0000market value is from dividends in the next year, implying a duration of 46\u0000years. What drives valuation duration? We find that market participants have\u0000limited information about cash flow beyond one year. Therefore, an increase in\u0000valuation duration is due to a decrease in the discount rate rather than good\u0000news about long-term growth. Accordingly, valuation duration negatively\u0000predicts annual market return with an out-of-sample R2 of 15%, robustly\u0000outperforming other predictors in the literature. While the price-dividend\u0000ratio reflects the overall valuation level, our valuation-based measure of\u0000duration captures the slope of the valuation term structure. We show that\u0000valuation duration, as a discount rate proxy, is a critical state variable that\u0000augments the price-dividend ratio in spanning the (latent) state space for\u0000stock-market dynamics.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522487","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper develops a coupled model for day-ahead electricity prices and average daily temperature which allows to model quanto weather and energy derivatives. These products have gained on popularity as they enable to hedge against both volumetric and price risks. Electricity day-ahead prices and average daily temperatures are modelled through non homogeneous Ornstein-Uhlenbeck processes driven by a Brownian motion and a Normal Inverse Gaussian L'evy process, which allows to include dependence between them. A Conditional Least Square method is developed to estimate the different parameters of the model and used on real data. Then, explicit and semi-explicit formulas are obtained for derivatives including quanto options and compared with Monte Carlo simulations. Last, we develop explicit formulas to hedge statically single and double sided quanto options by a portfolio of electricity options and temperature options (CDD or HDD).
{"title":"Risk valuation of quanto derivatives on temperature and electricity","authors":"Aurélien Alfonsi, Nerea Vadillo","doi":"arxiv-2310.07692","DOIUrl":"https://doi.org/arxiv-2310.07692","url":null,"abstract":"This paper develops a coupled model for day-ahead electricity prices and\u0000average daily temperature which allows to model quanto weather and energy\u0000derivatives. These products have gained on popularity as they enable to hedge\u0000against both volumetric and price risks. Electricity day-ahead prices and\u0000average daily temperatures are modelled through non homogeneous\u0000Ornstein-Uhlenbeck processes driven by a Brownian motion and a Normal Inverse\u0000Gaussian L'evy process, which allows to include dependence between them. A\u0000Conditional Least Square method is developed to estimate the different\u0000parameters of the model and used on real data. Then, explicit and semi-explicit\u0000formulas are obtained for derivatives including quanto options and compared\u0000with Monte Carlo simulations. Last, we develop explicit formulas to hedge\u0000statically single and double sided quanto options by a portfolio of electricity\u0000options and temperature options (CDD or HDD).","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522561","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Following the EU's decision to ban neonicotinoids, this article investigates the impacts of yellow virus on sugar beet yields under the ban and under current and future climates. Using a model that factors in key variables such as sowing dates, phenological stages, first aphid flight and aphid abundance, simulations are performed using long-period climate datasets as inputs. Coupled with incidence and sugar yield loss assumptions, this model allows to reconstruct the impact of yellow virus on sugar beet yields using a so called 'as if' approach. By simulating the effects of viruses over a longer period of time, as if neonicotinoids weren't used in the past, this methodology allows an accurate assessment of risks associated with yellow viruses, as well as impact of future agroecological mesures. The study eventually provides an actuarial rating for an insurance policy that compensates the losses triggered by those viruses.
{"title":"Actuarial Implications and Modeling of Yellow Virus on Sugar Beet After the EU's Ban on Neonicotinoids and Climate Change","authors":"Martial Phélippé-GuinvarcGAINS, Jean Cordier","doi":"arxiv-2310.01869","DOIUrl":"https://doi.org/arxiv-2310.01869","url":null,"abstract":"Following the EU's decision to ban neonicotinoids, this article investigates\u0000the impacts of yellow virus on sugar beet yields under the ban and under\u0000current and future climates. Using a model that factors in key variables such\u0000as sowing dates, phenological stages, first aphid flight and aphid abundance,\u0000simulations are performed using long-period climate datasets as inputs. Coupled\u0000with incidence and sugar yield loss assumptions, this model allows to\u0000reconstruct the impact of yellow virus on sugar beet yields using a so called\u0000'as if' approach. By simulating the effects of viruses over a longer period of\u0000time, as if neonicotinoids weren't used in the past, this methodology allows an\u0000accurate assessment of risks associated with yellow viruses, as well as impact\u0000of future agroecological mesures. The study eventually provides an actuarial\u0000rating for an insurance policy that compensates the losses triggered by those\u0000viruses.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper similar to [P. Carr, A. Itkin, 2019] we construct another Markovian approximation of the rough Heston-like volatility model - the ADO-Heston model. The characteristic function (CF) of the model is derived under both risk-neutral and real measures which is an unsteady three-dimensional PDE with some coefficients being functions of the time $t$ and the Hurst exponent $H$. To replicate known behavior of the market implied skew we proceed with a wise choice of the market price of risk, and then find a closed form expression for the CF of the log-price and the ATM implied skew. Based on the provided example, we claim that the ADO-Heston model (which is a pure diffusion model but with a stochastic mean-reversion speed of the variance process, or a Markovian approximation of the rough Heston model) is able (approximately) to reproduce the known behavior of the vanilla implied skew at small $T$. We conclude that the behavior of our implied volatility skew curve ${cal S}(T) propto a(H) T^{bcdot (H-1/2)}, , b = const$, is not exactly same as in rough volatility models since $b ne 1$, but seems to be close enough for all practical values of $T$. Thus, the proposed Markovian model is able to replicate some properties of the corresponding rough volatility model. Similar analysis is provided for the forward starting options where we found that the ATM implied skew for the forward starting options can blow-up for any $s > t$ when $T to s$. This result, however, contradicts to the observation of [E. Alos, D.G. Lorite, 2021] that Markovian approximation is not able to catch this behavior, so remains the question on which one is closer to reality.
{"title":"The ATM implied skew in the ADO-Heston model","authors":"Andrey Itkin","doi":"arxiv-2309.15044","DOIUrl":"https://doi.org/arxiv-2309.15044","url":null,"abstract":"In this paper similar to [P. Carr, A. Itkin, 2019] we construct another\u0000Markovian approximation of the rough Heston-like volatility model - the\u0000ADO-Heston model. The characteristic function (CF) of the model is derived\u0000under both risk-neutral and real measures which is an unsteady\u0000three-dimensional PDE with some coefficients being functions of the time $t$\u0000and the Hurst exponent $H$. To replicate known behavior of the market implied\u0000skew we proceed with a wise choice of the market price of risk, and then find a\u0000closed form expression for the CF of the log-price and the ATM implied skew.\u0000Based on the provided example, we claim that the ADO-Heston model (which is a\u0000pure diffusion model but with a stochastic mean-reversion speed of the variance\u0000process, or a Markovian approximation of the rough Heston model) is able\u0000(approximately) to reproduce the known behavior of the vanilla implied skew at\u0000small $T$. We conclude that the behavior of our implied volatility skew curve\u0000${cal S}(T) propto a(H) T^{bcdot (H-1/2)}, , b = const$, is not exactly\u0000same as in rough volatility models since $b ne 1$, but seems to be close\u0000enough for all practical values of $T$. Thus, the proposed Markovian model is\u0000able to replicate some properties of the corresponding rough volatility model.\u0000Similar analysis is provided for the forward starting options where we found\u0000that the ATM implied skew for the forward starting options can blow-up for any\u0000$s > t$ when $T to s$. This result, however, contradicts to the observation of\u0000[E. Alos, D.G. Lorite, 2021] that Markovian approximation is not able to catch\u0000this behavior, so remains the question on which one is closer to reality.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we propose an empirical model for the VIX index. Our findings indicate that the VIX has a long-term empirical distribution. To model its dynamics, we utilize a continuous-time Markov process with a uniform distribution as its invariant distribution and a suitable function $h$. We determined that $h$ is the inverse function of the VIX data's empirical distribution. Additionally, we use the method of variables of separation to get the exact solution to the pricing problem for VIX futures and call options.
{"title":"A Markovian empirical model for the VIX index and the pricing of the corresponding derivatives","authors":"Ying-Li Wang, Cheng-Long Xu, Ping He","doi":"arxiv-2309.08175","DOIUrl":"https://doi.org/arxiv-2309.08175","url":null,"abstract":"In this paper, we propose an empirical model for the VIX index. Our findings\u0000indicate that the VIX has a long-term empirical distribution. To model its\u0000dynamics, we utilize a continuous-time Markov process with a uniform\u0000distribution as its invariant distribution and a suitable function $h$. We\u0000determined that $h$ is the inverse function of the VIX data's empirical\u0000distribution. Additionally, we use the method of variables of separation to get\u0000the exact solution to the pricing problem for VIX futures and call options.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522568","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Stochastic volatility models, where the volatility is a stochastic process, can capture most of the essential stylized facts of implied volatility surfaces and give more realistic dynamics of the volatility smile or skew. However, they come with the significant issue that they take too long to calibrate. Alternative calibration methods based on Deep Learning (DL) techniques have been recently used to build fast and accurate solutions to the calibration problem. Huge and Savine developed a Differential Deep Learning (DDL) approach, where Machine Learning models are trained on samples of not only features and labels but also differentials of labels to features. The present work aims to apply the DDL technique to price vanilla European options (i.e. the calibration instruments), more specifically, puts when the underlying asset follows a Heston model and then calibrate the model on the trained network. DDL allows for fast training and accurate pricing. The trained neural network dramatically reduces Heston calibration's computation time. In this work, we also introduce different regularisation techniques, and we apply them notably in the case of the DDL. We compare their performance in reducing overfitting and improving the generalisation error. The DDL performance is also compared to the classical DL (without differentiation) one in the case of Feed-Forward Neural Networks. We show that the DDL outperforms the DL.
{"title":"Applying Deep Learning to Calibrate Stochastic Volatility Models","authors":"Abir Sridi, Paul Bilokon","doi":"arxiv-2309.07843","DOIUrl":"https://doi.org/arxiv-2309.07843","url":null,"abstract":"Stochastic volatility models, where the volatility is a stochastic process,\u0000can capture most of the essential stylized facts of implied volatility surfaces\u0000and give more realistic dynamics of the volatility smile or skew. However, they\u0000come with the significant issue that they take too long to calibrate. Alternative calibration methods based on Deep Learning (DL) techniques have\u0000been recently used to build fast and accurate solutions to the calibration\u0000problem. Huge and Savine developed a Differential Deep Learning (DDL) approach,\u0000where Machine Learning models are trained on samples of not only features and\u0000labels but also differentials of labels to features. The present work aims to\u0000apply the DDL technique to price vanilla European options (i.e. the calibration\u0000instruments), more specifically, puts when the underlying asset follows a\u0000Heston model and then calibrate the model on the trained network. DDL allows\u0000for fast training and accurate pricing. The trained neural network dramatically\u0000reduces Heston calibration's computation time. In this work, we also introduce different regularisation techniques, and we\u0000apply them notably in the case of the DDL. We compare their performance in\u0000reducing overfitting and improving the generalisation error. The DDL\u0000performance is also compared to the classical DL (without differentiation) one\u0000in the case of Feed-Forward Neural Networks. We show that the DDL outperforms\u0000the DL.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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